Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations course (despite the geometry) and wanted to learn some ODEs. What book would you recommend? This student would be happy to learn more analysis if necessary to understand what's in this ODE book.

In other words: I'm asking for your recommendations for a ODE book that is allowed to have arbitrary prerequisites from analysis and algebra and topology and even geometry, but with no knowledge of differential equations presumed.

Thank you!

(Note: It could have occurred to the hypothetical student to talk to his/her advisor/other faculty members, but in that case the student would still be interested in MathOverflow's response.)

It would help to know why you want such a book (e.g. as a prerequisite for diff. geometry?); until that is known, I recommend an undergraduate text, such as one by Martin Braun, that has applications as well as come coverage of theory. Gerhard "Ask Me About Motivating Remarks" Paseman, 2011.11.17
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Gerhard PasemanNov 18 '11 at 5:42

6 Answers
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There are way too many approaches to ODEs to have any one book cover them all. I occasionally use a book called Differential Equations and Dynamical Systems, by Lawrence Perko. The focus of this book is on qualitative behavior - existence of fixed points, limit cycles, blow-up solutions, etc.

I would not call this a standard introduction to ODE - it does not cover some of the absolute basics. However, I think the emphasis of this text on geometry, and on using some more modern results, makes the book a decent choice.

Some flaws: The book really only presupposes mastery of analysis. There are some tools missing, in particular from geometry/topology, that could make the presentation a bit cleaner. It sounds like you have a strong geometry/topology background, so maybe this disqualifies this text for you.

For a more classical treatment of ODEs, in particular the treatment of ODEs as linear operators (Sturm-Liouville theory), I might go for Coddington's Theory of Ordinary Differential Equations. It is very classical, but it really does cover all the essential theory.

Henri Cartan's course in differential calculus does cover quite a few useful things for differential equations, from a high-level point of view : you'll find the notion of differentiation in a generic form, the big theorems are proven (local inversion, Cauchy-Lipschitz, ...).

For the low-level and the explicit, Arnold's "Ordinary differential equations" is a must-read, as Qiaochu Yuan already pointed out.

I really like Ordinary Differential Equations by Jack K. Hale. It's very rigorous
and thorough in the fundamentals, has a great section on periodic linear systems, and covers
some advanced stuff such as integral manifolds. Arnold, Abraham and Marsden, and Hirsch, Smale and Devaney are also nice, though the emphasis is different.